The theory of the covariant Dirac field defined on \(M\)  was initiated by Nachtmann [1]} and developed then by one of us [2]}, [3]}, [4]}, [5]}, [6]}, [7]}. In this approach the Dirac field transforms under continuous \(O(1,4)\)  isometries according to the Dirac representation \(\rho _D\)  of the \(SL(2,{C})={\rm Spin}L_+^{\uparrow }\)  group which is the universal covering group of the proper orthochronus Lorentz group \(L_+^{\uparrow }\)  that is gauge group of \(M\)  [8]}. The \(SL(2,{C})\)  transformations are associated through the canonical homomorphism to the gauge ones which have to correct the tetrad gauge after each isometry in order to leave the Dirac equation invariant [2]} but giving rice to the associated conserved quantities predicted by Carter and McLenaghan [10]}, [11]}, [12]}. For this reason we say that the covariant representation of the isometry group \(SO(1,4)\)  is induced by \(\rho _D\) . In other words this approach extends the Lorentz covariance of special relativity to general relativity. This framework offers many attractive applications [13]}, [14]}, [15]}, [16]}, [17]}, [18]} such that the study of the discrete de Sitter isometries generating the discrete symmetries of the Dirac theory was neglected so far. For this reason we devote this paper to the discrete transformations of the covariant Dirac field on the de Sitter manifold.
